Historically, the idea to investigate the difference (or sum) of the values of a holomorphic function with a prescribed sign of its imaginary or real part dates back to the 1910's. A fundamental result from this era is due to Pick [1]}, who proved, in modern terms, that for any holomorphic function \(f\colon  with non-negative real part it holds that\)\(\frac{f(\xi ) + \overline{f(\eta )}}{1-\xi \,\overline{\eta }} = \frac{1}{\pi }\int _{[0,2\pi )}\frac{1}{({\mathrm {e}}^{\mathfrak {i}\,s}-\xi )({\mathrm {e}}^{-\mathfrak {i}\,s}-\overline{\eta })}\mathrm {d}\nu (s),\)\(where \) , and \(\nu \)  is the representing measure of the function \(f\)  in the sense of the Riesz-Herglotz representation theorem [2]}. Using the Cayley transform and its inverse, recalled explicitly later on, this result may be reinterpreted in the language of Herglotz-Nevanlinna functions where it reproduces the results summarized in the beginning of Section .
